We describe the canonical correspondence between finite metric spaces and
symmetric convex polytopes, and formulate the problem about
classification of the metric spaces in terms of combinatorial
structure of those polytopes.

We will discuss problems with very
elementary formulations that concern the most popular notions in
mathematics: metric spaces, convex geometry, combinatorics of
polytopes and Kantorovich’s optimal transportation. According to
Arnold’s classification, there are two ways to introduce a new
subject: the first way (he called it the “Russian tradition”) is
to start with “the simplest non-trivial partial case”—I will use
this approach. The second and the opposite tradition, which I also
like very much (he called it “Bourbaki’s tradition”) is to start
with an “extremely general case that is impossible to further
generalize”.

So my metric spaces will be finite, polytopes will be
finite-dimensional etc. but all the notions and problems have
infinite, infinite-dimensional, and continuous analogs.

General Problem

Study and classify finite metric spaces according to combinatorial properties of their fundamental
polytopes, associated with metric spaces in a canonical fashion. A
more precise formulation follows.

In the rest of the paper, I explain the terminology that is used in
the title and in the problem.

Let $(X,\rho)$, $|X|=n$, be a finite metric space. We will write
$V(X)$ for the vector space of all real-valued functions on $X$. The
value of a function $v\in V(X)$ at a point $x\in X$ will be denoted
by $v^{x}$. The space $V(X)$ can be naturally identified with the
space of all formal linear combinations of elements of $X$. Under
this identification, any element $x\in X$ identifies with the
delta-function equal to 1 at $x$ and to 0 at all other points of
$X$. Define the real vector space $V_{0}(X)$ as a subspace of $V(X)$
consisting of all $v\in V(X)$ with $\sum_{x\in X}v^{x}=0$.

Let us define the map $\delta:X\to V_{0}(X)$ taking an element $x$ to
the function $\delta(x)=\delta_{x}\in V_{0}(X)$ such that
$\delta_{x}^{y}=-\frac{1}{n}$ for all $y\neq x$. Then we must have
$\delta_{x}^{x}=\frac{n-1}{n}$.

The convex hull $\mathrm{Conv}\left[\{\delta(x)\},x\in X\right]$ of the image
of this map is a $(n-1)$-dimensional simplex denoted by $\Sigma(X)\subset V_{0}(X)$ (this simplex is obtained from the standard
coordinate simplex in $V(X)$ by the projection mapping $x\in X$ to
$\delta(x)$).

The metric $\rho$ can be considered now as a metric on the vertices
of the simplex $\Sigma(X)$, we use the same symbol $\rho$ to denote
this metric. For every pair of distinct points $x$, $y\in X$,
consider the vectors $e_{x,y}\in V_{0}$ defined by the formula:

$e_{x,y}=\frac{\delta_{x}-\delta_{y}}{\rho(x,y)}.$

This vectors will play a major role in what follows.

Definition 1.

The fundamental polytope of a metric space $(X,\rho)$ is
the convex polytope $R_{X,\rho}$ obtained as the convex hull of all
vectors $e_{x,y}$, where $(x,y)$ run through all pairs of distinct
points of $X$. The combinatorial type of $R_{X,\rho}$, i.e., the
isomorphism class of the corresponding poset of faces, is called the
combinatorial structure of the finite metric space
$(X,\rho)$. Two finite metric spaces with the same combinatorial
structures called similar metric spaces.

It is easy to see that the fundamental polytope $R_{X,\rho}$ is
centrally symmetric (i.e., it coincides with its reflection in the
origin). If $\rho(x,y)=1$ for any pair of distinct elements $x$,
$y\in X$, then the fundamental polytope is the Minkowski sum of two
simplices $\Sigma(X)$ and $-\Sigma(X)$.

In general, we consider the fixed set of rays $\left\{\lambda(\delta_{x}-\delta_{y}):\lambda>0\right\}$, which is independent of the metric, and
choose one point in each ray with $\lambda=\frac{1}{\rho(x,y)}$
(this choice now depends on the metric). The fundamental polytope is
the convex hull of all thus obtained points.

A simple fact (Mulleray et al.2008) that characterizes the fundamental
polytopes of finite metric spaces, is the following: given a
symmetric function $\rho:X\times X\to\mathbb{R}_{>0}$, consider the
polytope

Then this function $\rho$ is a metric on $X$ if and only if no point
$e_{x,y}$ belongs to the interior of convex hull of the other
points.

The polytope $R_{X,\rho}$ is convex and central symmetric;
therefore, it defines the Minkowski–Banach norm $\|\cdot\|_{\rho}$
in the real vector space $V_{0}(X)$, whose unit ball is by definition
the polytope $R_{X,\rho}$. In the finite-dimensional case, this norm
is the so called Kantorovich–Rubinstein norm. If
$\rho(x,y)=1$ whenever $x\neq y$, then the corresponding fundamental
polytope is the so called root polytope, and the
corresponding Kantorovich–Rubinstein norm is the restriction to
$V_{0}(X)$ of the $\ell^{1}$-norm in the space $V(X)$.

Thus we reduce the analysis of metric space to the convex geometry
of fundamental polytope. Since the space $X$ is isometrically
embedded into $V_{0}(X)$ (points of $X$ correspond to the vertices of
the simplex $\Sigma(X)$), endowed with metric $\rho$ (see above)
we can restore metric on $X$ using fundamental polytope.

Recall that the combinatorial type of a convex polytope is the
isomorphism class of the partial ordered set form by the faces of
the polytope, ordered by inclusion; the $f$-vector of a convex
polytope is the finite tuple $(f_{0},f_{1},\ldots f_{n})$, where $f_{0}=n$
is the number of vertices, $f_{1}$ is the number of edges, etc.,
$f_{n-1}$ is number of facets (i.e., faces of codimension 1) and,
finally, $f_{n}=1$.

Our definition is functorial in the sense that each isometry of one
metric space to another vector space produces an affine isomorphism
of the corresponding fundamental polytopes.

We may say that the notion of the combinatorial type of metric
spaces defines a natural stratification of the cone ${\mathcal{M}}_{n}$
of all distance matrices (i.e., real symmetric $n\times n$ matrices,
whose entries are the values of a distance function defined on some
finite metric space of cardinality $n$). Below we suggest to study
this important stratification, more precisely, to solve the
following problems.

Problem 1.

1.

Express the combinatorial structure of $(X,\rho)$ in terms of the
metric $\rho$, i.e., find the $f$-vector of the corresponding
fundamental polytope in terms of the metric $\rho$ itself—using
linear inequalities on the values of metric etc.

2.

Estimate the number of combinatorial structures for any given $n$
and study its asymptotic behavior as $n$ tends to infinity. The most
interesting thing is to estimate the number of “open” (generic)
types.

3.

Provide sufficient conditions on two metric spaces to be similar.

It is well known that most finite metric spaces cannot be
isometrically imbedded into a Euclidean or a Hilbert space (we say
that these metrics do not have Euclidean type). The following
question appears:

4.

Describe the combinatorial types of metric spaces of Euclidean
type. Do we obtain all combinatorial types?

5.

Does the stratification of the space of distance matrices into
the combinatorial types is universal? Or, on the contrary, there are some restrictions on the topological types of the open
components?11A classification problem (in algebraic
geometry, combinatorics, etc.) of a certain set of objects up to a
certain equivalence relation is called a “universal problem” if
all possible kinds of singularities or stratifications, say, of
arbitrary semi-algebraic varieties can occur in the topology of
equivalence classes.
A well-known example is Mnev’s theorem on the universality of the
combinatorial classification of convex polytopes in dimensions $\geq 4$. (see the papers by (Vershik and Mnev1988) and more recent literature).

Consider a very degenerate metric space with $n$ points:
$X=\textbf{n}$, the set of all integers from 1 to $n$ with mutual
distances between all pairs of distinct points equal to 1:
$\rho(i,j)=\delta_{i,j}$.

In this case, the simplex $\Sigma(X)$ is a regular Euclidean
simplex. We can view it as $(n-1)$-simplex in a Cartan subalgebra of
the Lie algebra $A_{n}$. From this viewpoint, the simplex is spanned
by all positive simple roots $e_{i,i+1}$, $i=1$, $\dots$, $n-1$, and
one maximal root $e_{n,1}$.

Proposition 1.

Let $X=\textbf{n}$ and $\rho(i,j)=\delta_{i,j}$, $i,j=1$, $\dots$,
$n$. Identify the vector space $V_{0}(X)$ with a Cartan subalgebra of
the Lie algebra $A_{n}$. Then the fundamental polytope $R_{(X,\rho)}$
is the convex hull of all roots. The norm $\|.\|_{\rho}$ associated
with the fundamental polytope coincides with the restriction of the
$\ell^{1}$-norm on $V(X)$

to the hyperplane $V_{0}(X)\subset V(X)$. Thus, the polytope
$R_{X,\rho}$ in this case is the intersection of an $n$-dimensional
octahedron with the hyperplane $V_{0}(X)$.

The corresponding norm $\|\cdot\|$ on the Lie algebra of
skew-hermitian matrices with zero trace is the “nuclear norm”, for
which the norm of a matrix is the sum of the moduli of all its
eigenvalues.

It is natural to call $R_{X,\rho}$ the “root polytope”. An easy
exercise is to find the $f$-vector in this case. For example, if
$n=3$ and $\dim(V_{0})=2$, then the fundamental polygon is a regular
hexagon, and the norm is the hexagonal norm. For $n=4$, see the
Fig. 1: the $f$-vector is equal to $(12,24,14)$, the facets are 8
regular triangles and 6 squares. For $n=3$, the group of symmetries
of the fundamental polytope is the dihedral group
$D_{6}={\mathbb{Z}}_{2}\rightthreetimes{\mathbb{Z}}_{6}$. For $n=4$,
see the Fig. 2.

Note that the group of symmetries of the root polytope includes the
Weyl group (which is isomorphic to the symmetric group). Root
polytopes were mentioned for other reason in (Gelfand and Kapranov1993).

Why is the definition of the fundamental polytope $R_{X,\rho}$
associated with a finite metric space $(X,\rho)$ natural? The
justification is as follows. We want to define a natural metric
$\bar{\rho}$ on $V_{0}(X)$ that extends the metric $\rho$ on $X$. The
latter is identified with the set of vertices of the simplex
$\Sigma(X)$ so that the distance between two points $x$ and $y$ in
$X$ coincides with the distance between the corresponding vertices
$v_{x}$ and $v_{y}$ of the simplex $\Sigma(X)$. In another words, we
want to define a norm $\|\cdot\|$ in $V_{0}(X)$ with the property
$\|e_{x,y}\|=\rho(x,y)$. There are many such extensions but there is
a maximal one:

Theorem 1.

(Mulleray et al.2008) The norm $\|\cdot\|_{\rho}$, called the
Kantorovich–Rubinstein norm, is the unique maximal extension of
$\rho$: all other norms possessing this extension property are less
that this one because the fundamental polytope is contained in the
unit balls of all such norms.

,

The direct description of an extension of the metric $\rho$ to the
whole simplex $\Sigma(X)$ is a classical definition by Kantorovich
of his transportation metric. This definition is well known in the
mathematical economics and in linear programming but there were no
publications [before (Mulleray et al.2008), see comments below], in which
characteristic properties of fundamental polytopes are discussed and
serious combinatorial investigations are conducted. Recall, for the
sake of completeness, the classical definition of the
Kantorovich transportation metric, which is equivalent to our
definition above.

Consider the positive part $v(+)$ and the negative part $v(-)$ of
the vector $v$. The vector $v(+)$ is defined as the componentwise
maximum of $v$ and the zero vector, and the vector $v(-)$ is defined
as the componentwise minimum. Evidently, we have $v=v(+)+v(-)$. Thus
we have two nonnegative vectors $v(+)=u$ and $v(-)=w$ with equal
sums of coordinates.

Here $u^{x}$ is the coordinate of the vector $u$ corresponding to the
point $x\in X$, and similarly for $w$.

For the history of the Kantorovich metric, see (Vershik2013) and
references therein. For some further development and applications of
the finite-dimensional geometry of this metric, see (Vershik2014).

The last question is also of Arnold’s style (see e.g. (Arnold2006)): in
our definition of the Kantorovich metric, we used only the root
system $A_{n}$. My question is: what are the analogs of this
definition (and maybe even of the transportation problem!) for other
series of Lie algebras and other Weyl groups.